Using the technique of [1]},
let \(R=\left(\frac{2b}{a}\right)^{\frac{1}{\beta }}\) , we lower bound
\(U\left(x\right)\)  when \(\left\Vert x\right\Vert \le R\) ,
\(U(x) & =U(0)+\int _{0}^{1}\left\langle \nabla U(tx),\ x\right\rangle dt\nonumber \\& \ge U(0)-\int _{0}^{1}\left\Vert \nabla U(tx)\right\Vert \left\Vert x\right\Vert dt\nonumber \\& \ge U(0)-\sum _{i}L_{i}\left\Vert x\right\Vert ^{\alpha _{i}+1}\int _{0}^{1}t^{\alpha _{i}}dt\nonumber \\& \ge U(0)-\sum _{i}\frac{L_{i}}{\alpha _{i}+1}\left\Vert x\right\Vert ^{\alpha _{i}+1}\nonumber \\& \ge U(0)-\sum _{i}\frac{L_{i}}{\alpha _{i}+1}R^{\alpha _{i}+1}\) 
